This module discusses about types of bonds and theories of Bond Valuations

1. Valuation of Bonds and
Shares
Dr. Vinita Kalra

2. 2
Introduction
• Assets can be real or financial; securities
like shares and bonds are called financial
assets while physical assets like plant and
machinery are called real assets.
• The concepts of return and risk, as the
determinants of value, are as fundamental
and valid to the valuation of securities as to
that of physical assets.

3. 3
Concept of Value
• Book Value
• Replacement Value
• Liquidation Value
• Going Concern Value
• Market Value

4. 4
Features of a Bond
• Face Value
• Interest Rate—fixed or floating
• Maturity
• Redemption value
• Market Value

5. 5
Bonds Values and Yields
• Bonds with maturity
• Pure discount bonds
• Perpetual bonds

6. 6
Bond with Maturity
Bond value = Present value of interest + Present
value of maturity value:
0
1
INT
(1 ) (1 )
n
t n
t n
t d d
B
B
k k
 
 


7. 7
Yield to Maturity
• The yield-to-maturity (YTM) is the measure
of a bond’s rate of return that considers
both the interest income and any capital
gain or loss. YTM is bond’s internal rate of
return.
• A perpetual bond’s yield-to-maturity:
0
1
INT INT
(1 )
n
t
t d d
B
k k


 



8. 8
Current Yield
• Current yield is the annual interest divided
by the bond’s current value.
• Example: The annual interest is Rs 60 on
the current investment of Rs 883.40.
Therefore, the current rate of return or the
current yield is: 60/883.40 = 6.8 per cent.
• Current yield does not account for the
capital gain or loss.

9. 9
Yield to Call
• For calculating the yield to call, the call period
would be different from the maturity period
and the call (or redemption) value could be
different from the maturity value.
• Example: Suppose the 10% 10-year Rs 1,000
bond is redeemable (callable) in 5 years at a
call price of Rs 1,050. The bond is currently
selling for Rs 950.The bond’s yield to call is
12.7%.
   
5
5
1
100 1,050
950
1 YTC 1 YTC
t
t
 
 


10. 10
Bond Value and Amortisation of
Principal
• A bond (debenture) may be amortised every
year, i.e., repayment of principal every year
rather at maturity.
• The formula for determining the value of a
bond or debenture that is amortised every
year, can be written as follows:
– Note that cash flow, CF, includes both the interest and
repayment of the principal.
0
1 (1 )
n
t
t
t d
CF
B
k




11. 11
Pure Discount Bonds
• Pure discount bond do not carry an explicit
rate of interest. It provides for the payment of
a lump sum amount at a future date in
exchange for the current price of the bond.
The difference between the face value of the
bond and its purchase price gives the return
or YTM to the investor.

12. 12
Pure Discount Bonds
• Example: A company may issue a pure
discount bond of Rs 1,000 face value for
Rs 520 today for a period of five years.
The rate of interest can be calculated as
follows:
 
 
5
5
1/5
1,000
520
1 YTM
1,000
1 YTM 1.9231
520
1.9231 1 0.14 or 14%i


  
  

13. 13
Pure Discount Bonds
• Pure discount bonds are called deep-
discount bonds or zero-interest bonds or
zero-coupon bonds.
• The market interest rate, also called the
market yield, is used as the discount rate.
• Value of a pure discount bond = PV of the
amount on maturity:
 
0
1
n
n
d
M
B
k



14. 14
Perpetual Bonds
• Perpetual bonds, also called consols, has an
indefinite life and therefore, it has no
maturity value. Perpetual bonds or
debentures are rarely found in practice.

15. 15
Perpetual Bonds
• Suppose that a 10 per cent Rs 1,000 bond will
pay Rs 100 annual interest into perpetuity.
What would be its value of the bond if the
market yield or interest rate were 15 per cent?
• The value of the bond is determined as
follows:
0
INT 100
Rs 667
0.15d
B
k
  

16. 16
Bond Values and Changes in Interest
Rates
• The value of the bond
declines as the market
interest rate (discount
rate) increases.
• The value of a 10-year, 12
per cent Rs 1,000 bond
for the market interest
rates ranging from 0 per
cent to 30 per cent. 0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
0% 5% 10% 15% 20% 25% 30%
Interest Rate
BondValue

17. 17
Bond Maturity and Interest Rate Risk
• The intensity of interest rate
risk would be higher on bonds
with long maturities than
bonds with short maturities.
• The differential value response
to interest rates changes
between short and long-term
bonds will always be true.
Thus, two bonds of same
quality (in terms of the risk of
default) would have different
exposure to interest rate risk.
PresentValue(Rs)
Discountrate(%) 5-Yearbond 10-Yearbond Perpetualbond
5 1,216 1,386 2,000
10 1,000 1,000 1,000
15 832 749 667
20 701 581 500
25 597 464 400
30 513 382 333

18. 18
Bond Maturity and Interest Rate Risk

19. 19
Bond Duration and Interest Rate
Sensitivity
• The longer the maturity of a bond, the higher
will be its sensitivity to the interest rate
changes. Similarly, the price of a bond with
low coupon rate will be more sensitive to the
interest rate changes.
• However, the bond’s price sensitivity can be
more accurately estimated by its duration. A
bond’s duration is measured as the weighted
average of times to each cash flow (interest
payment or repayment of principal).

20. 20
Duration of Bonds
• Let us consider the 8.5
per cent rate bond of Rs
1,000 face value that
has a current market
value of Rs 954.74 and a
YTM of 10 per cent, and
the 12 per cent rate
bond of Rs 1,000 face
value has a current
market value of Rs
1,044.57 and a yield to
maturity of 10.8 per
cent. Table shows the
calculation of duration
for the two bonds.
8.5 Percent Bond
Year Cash Flow
Present Value
at 10 %
Proportion of
Bond Price
Proportion of
Bond Price x Time
1 85 77.27 0.082 0.082
2 85 70.25 0.074 0.149
3 85 63.86 0.068 0.203
4 85 58.06 0.062 0.246
5 1,085 673.70 0.714 3.572
943.14 1.000 4.252
11.5 Percent Bond
Year
Cash
Flow
Present Value
at 10.2%
Proportion of
Bond Price
Proportion of Bond
Price x Time
1 115 103.98 0.101 0.101
2 115 94.01 0.091 0.182
3 115 85.00 0.082 0.247
4 115 76.86 0.074 0.297
5 1,115 673.75 0.652 3.259
1,033.60 1.000 4.086

21. 21
Volatility
• The volatility or the interest rate sensitivity of a bond is
given by its duration and YTM. A bond’s volatility, referred
to as its modified duration, is given as follows:
• The volatilities of the 8.5 per cent and 11.5 per cent bonds
are as follows:
Duration
Volatility of a bond
(1 YTM)


4.086
Volatility of 11.5% bond 3.69
(1.106)
 
4.252
Volatility of 8.5% bond 3.87
(1.100)
 

22. 22
The Term Structure of Interest Rates
• Yield curve shows the relationship between the yields to
maturity of bonds and their maturities. It is also called the
term structure of interest rates.
• Yield Curve (Government of India Bonds)
5.90%
7.18%
5.0%
5.5%
6.0%
6.5%
7.0%
7.5%
0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 >10
Maturity
(Years)
Yield (%)

23. 23
The Term Structure of Interest Rates
• The upward sloping yield curve implies that
the long-term yields are higher than the
short-term yields. This is the normal shape
of the yield curve, which is generally
verified by historical evidence.
• However, many economies in high-inflation
periods have witnessed the short-term
yields being higher than the long-term
yields. The inverted yield curves result
when the short-term rates are higher than
the long-term rates.

24. 24
The Expectation Theory
• The expectation theory supports the
upward sloping yield curve since investors
always expect the short-term rates to
increase in the future.
• This implies that the long-term rates will be
higher than the short-term rates.
• But in the present value terms, the return
from investing in a long-term security will
equal to the return from investing in a
series of a short-term security.

25. 25
The Expectation Theory
• The expectation theory assumes
– capital markets are efficient
– there are no transaction costs and
– investors’ sole purpose is to maximize their returns
• The long-term rates are geometric average of current and
expected short-term rates.
• A significant implication of the expectation theory is that
given their investment horizon, investors will earn the
same average expected returns on all maturity
combinations.
• Hence, a firm will not be able to lower its interest cost in
the long-run by the maturity structure of its debt.

26. 26
The Liquidity Premium Theory
• Long-term bonds are more sensitive than the
prices of the short-term bonds to the changes in
the market rates of interest.
• Hence, investors prefer short-term bonds to the
long-term bonds.
• The investors will be compensated for this risk
by offering higher returns on long-term bonds.
• This extra return, which is called liquidity
premium, gives the yield curve its upward bias.

27. 27
The Liquidity Premium Theory
• The liquidity premium theory means that
rates on long-term bonds will be higher
than on the short-term bonds.
• From a firm’s point of view, the liquidity
premium theory suggests that as the cost
of short-term debt is less, the firm could
minimize the cost of its borrowings by
continuously refinancing its short-term
debt rather taking on long-term debt.

28. 28
The Segmented Markets Theory
• The segmented markets theory assumes
that the debt market is divided into several
segments based on the maturity of debt.
• In each segment, the yield of debt depends
on the demand and supply.
• Investors’ preferences of each segment
arise because they want to match the
maturities of assets and liabilities to reduce
the susceptibility to interest rate changes.

29. 29
The Segmented Markets Theory
• The segmented markets theory approach
assumes investors do not shift from one
maturity to another in their borrowing—
lending activities and therefore, the shift in
yields are caused by changes in the demand
and supply for bonds of different maturities.

30. 30
Default Risk and Credit Rating
• Default risk is the risk that a company will
default on its promised obligations to
bondholders.
• Default premium is the spread between
the promised return on a corporate bond
and the return on a government bond with
same maturity.

31. 31
Crisil’s Debenture Ratings
High Investme nt Gr ades
AAA (Triple A): Highest Safety Debentures rated `AAA' are judged to offer highes t safety of
timely payment of interest and principal. Though the
circu mstances providing this degree of safety are like ly to
change, such changes as can be envisaged are most unlikely to
affect adversely the fundamentally strong position of such issues.
AA (Double A): High Safety Debentures rated 'AA' are judged to offer high safety of time ly
payment of interest and principal. They differ in safety fro m
`AAA' issues only margina lly.
Investment Gr ades
A: Adequate Safety Debentures rated `A' are judged to offer adequate safety of time ly
payment of interest and principal; however, changes in
circu mstances can adversely affect such issues more than those in
the higher rated categories.
BBB (T rip le B): Moderate Safety Debentures rated `BBB' are judged to offer sufficient safety of
timely payment of interest and principal for the present; however,
changing circumstances are more like ly to lead to a weakened
capacity to pay interest and repay principal than for debentures in
higher rated categories.
Speculati ve Gr ades
BB (Double B): Inadequate Safety Debentures rated `BB' are judged to carry inadequate safety of
timely pay ment of interest and principal; wh ile they are less
susceptible to default than other speculative grade debentures in
the immediate future, the uncertainties that the issuer faces could
lead to inadequate capacity to ma ke timely interest and principal
payments.
B: High Risk Debentures rated `B' are judged to have greater susceptibility to
default; while currently interest and principal payments are met,
adverse business or economic conditions would lead to lack of
ability or willingness to pay interest or principal.
C: Substantial Risk Debentures rated `C' are judged to have factors present that make
them vulnerable to default; time ly payment of interest and
principal is possible only if favourable c ircu mstances continue.
D: In De fault Debentures rated `B' are judged to have greater susceptibility to
default; while currently interest and principal payments are met,
adverse business or economic conditions would lead to lack of
ability or willingness to pay interest or principal.
Note:
1. CRISIL may apply " +" (plus) or " -" (minus) signs for ratings from AA to D to reflect comparative standing
within th e category.
2. The contents within parenth esis are a guide to the pronuncia tion of the rating symbo ls.
3. Preference share rating symbols are identical to deben ture rating symbols except that th e letters "pf" are
prefixed to the d ebenture rating symbols, e.g. pfAAA ("pf Triple A" ).

32. 32
Valuation of Shares
• A company may issue two types of shares:
– ordinary shares and
– preference shares
• Features of Preference and Ordinary Shares
– Claims
– Dividend
– Redemption
– Conversion

33. 33
Valuation of Preference Shares
• The value of the preference share would be
the sum of the present values of dividends
and the redemption value.
• A formula similar to the valuation of bond
can be used to value preference shares with
a maturity period:
1
0
1
PDIV
(1 ) (1 )
n
n
t n
t p p
P
P
k k
 
 


34. 34
Suppose an investor is considering the purchase of a 12-year, 10% Rs 100 par value preference share. The
redemption value of the preference share on maturity is Rs 120. The investor’s required rate of return is
10.5 percent. What should she be willing to pay for the share now? The investor would expect to receive
Rs 10 as preference dividend each year for 12 years and Rs 110 on maturity (i.e., at the end of 12 years).
We can use the present value annuity factor to value the constant stream of preference dividends and the
present value factor to value the redemption payment.
30.101Rs24.3606.65302.0120506.610
)105.1(
120
)105.1(105.0
1
105.0
1
10P 12120












Note that the present value of Rs 101.30 is a composite of the present value of dividends, Rs 65.06 and
the present value of the redemption value, Rs 36.24.The Rs 100 preference share is worth Rs 101.3 today
at 10.5 percent required rate of return. The investor would be better off by purchasing the share for Rs 100
today.
Value of a Preference Share-Example

35. 35
Valuation of Ordinary Shares
• The valuation of ordinary or equity shares is
relatively more difficult.
– The rate of dividend on equity shares is not known;
also, the payment of equity dividend is discretionary.
– The earnings and dividends on equity shares are
generally expected to grow, unlike the interest on
bonds and preference dividend.

36. 36
Dividend Capitalisation
• The value of an ordinary share is
determined by capitalising the future
dividend stream at the opportunity cost of
capital
• Single Period Valuation:
– If the share price is expected to grow at g per cent,
then P1:
– We obtain a simple formula for the share valuation as
follows:
1 1
0
DIV
1 e
P
P
k



1 0 (1 )P P g 
1
0
DIV
e
P
k g



37. 37
Multi-period Valuation
• If the final period is n, we can write the
general formula for share value as follows:
• Growth in Dividends
– Normal Growth
– Super-normal Growth
0
1
DIV
(1 ) (1 )
n
t n
t n
t e e
P
P
k k
 
 

Growth = Retention ratio Return on equity
ROEg b

 
1
0
DIV
e
P
k g


Share value PV of dividends during finite super-normal growth period
PV of dividends during indefinite normal growth period



38. 38
Earnings Capitalisation
• Under two cases, the value of the share can
be determined by capitalising the expected
earnings:
– When the firm pays out 100 per cent dividends; that is,
it does not retain any earnings.
– When the firm’s return on equity (ROE) is equal to its
opportunity cost of capital.

39. 39
Equity Capitalisation Rate
• For firms for which dividends are expected
to grow at a constant rate indefinitely and
the current market price is given
1
0
DIV
ek g
P
 

40. 40
Caution in Using Constant-Growth
Formula
• Estimation errors
• Unsustainable high current growth
• Errors in forecasting dividends

41. 41
Valuing Growth Opportunities
• The value of a growth opportunity is
given as follows:
1
1
NPV
EPS (ROE )
( )
g
e
e
e e
V
k g
b k
k k g


 



42. 42
Price-Earnings (P/E) Ratio: How
Significant?
• P/E ratio is calculated as the price of a share
divided by earning per share.
• Some people use P/E multiplier to value the
shares of companies.
• Alternatively, you could find the share value
by dividing EPS by E/P ratio, which is the
reciprocal of P/E ratio.

43. 43
Price-Earnings (P/E) Ratio: How
Significant?
• The share price is also given by the
following formula:
• The earnings price ratio can be derived as
follows:
1
0
EPS
g
e
P V
k
 
1EPS
1
g
e
o o
V
k
P P
 
  
 

44. 44
Price-Earnings (P/E) Ratio: How
Significant?
• Cautions:
– E/P ratio will be equal to the capitalisation rate only if
the value of growth opportunities is zero.
– A high P/E ratio is considered good but it could be
high not because the share price is high but because
the earnings per share are quite low.
– The interpretation of P/E ratio becomes meaningless
because of the measurement problems of EPS.